conical and glancing jahn-t eller intersections in the cyclic … · 2019. 5. 21. · conical and...

Conical and glancing Jahn-Teller intersections in the cyclic trinitrogencation

Vadim A. MozhayskiyDepartment of Chemistry, University of Southern California, Los Angeles, California 90089-0482

Dmitri BabikovChemistry Department, Marquette University, Milwaukee, Wisconsin 53201-1881

Anna I. Krylova!Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482

!Received 22 March 2006; accepted 19 April 2006; published online 12 June 2006"

The ground and electronically excited states of cyclic N3+ are characterized at the equilibrium D3h

geometry and along the Jahn-Teller distortions. Lowest excited states are derived from singleexcitations from the doubly degenerate highest occupied molecular orbitals !HOMOs" to the doublydegenerate lowest unoccupied molecular orbitals !LUMOs", which give rise to two exactly and twonearly degenerate states. The interaction of two degenerate states with two other states eliminateslinear terms and results in a glancing rather than conical Jahn-Teller intersection. HOMO-2→LUMOs excitations give rise to two regular Jahn-Teller states. Optimized structures, vertical andadiabatic excitation energies, frequencies, and ionization potential !IP" are presented. IP is estimatedto be 10.595 eV, in agreement with recent experiments. © 2006 American Institute of Physics.#DOI: 10.1063/1.2204602$

I. INTRODUCTION

Interest in homonuclear triatomic molecules has a longhistory. This is the smallest nonlinear system that can have anon-Abelian point group symmetry containing irreduciblerepresentations of the order higher than one, which results insymmetry required degeneracies between some electronicstates at high symmetry geometries !D3h or equilateral tri-angle" and the intersections between the corresponding po-tential energy surfaces !PESs".

According to the Jahn-Teller !JT" theorem,1–3 high sym-metry intersection points in nonlinear systems are not sta-tionary points on PESs, that is, degenerate states follow firstorder JT distortions to a lower symmetry, which lifts thedegeneracy. The linear dependence of states’ energies nearthe intersection gives rise to singularity points on adiabaticPESs. More precisely, the theorem states that for any geom-etry with symmetry required degeneracy between the elec-tronic states there exists a nuclear displacement along whichthe linear derivative coupling matrix element between theunperturbed states and the difference between the diagonalmatrix elements of the perturbation are not required to bezero by symmetry. The theorem does not guarantee the inter-section in real physical problems and the above vibronicterms can become zero due to additional symmetries, prop-erties of the potential, or other reasons. For example, as men-tioned in the original paper, the energy splitting can be neg-ligible if the orbital dependence on the displacement is weak,as in the case of lone pairs.

An interesting situation arises when such JT pair inter-acts with other closely lying states, as it happens in cyclic

N3+. In this case the overall energy dependence at the inter-

section point becomes purely quadratic and high symmetryconfigurations can become stationary points on adiabaticPESs.

In triatomics, degenerate PESs usually form a conicalintersection !CI" extensively characterized over the years fora variety of systems.4,5 An important feature of such inter-section is that the real electronic wave function calculatedwithin the Born-Oppenheimer approximation gains a phase,i.e., a sign change, along any path on the PES, which en-circles a conical intersection and thus contains the singularitypoint.6–9 This geometric phase has a profound effect on thenodal structure of the vibrational wave functions, the orderof vibrational states, and the selection rules for the vibra-tional transitions.10,11

A common motif in these systems is an unpaired elec-tron in one of the doubly degenerate molecular orbitals!MOs" in the ground electronic state. Alternatively, CIs canbe formed by the excited states derived from electron transi-tions between doubly degenerate and non-degenerate orbit-als, e.g., nondegenerate highest occupied molecular orbitial!HOMO" and doubly degenerate lowest unoccupied molecu-lar orbital !LUMO" in NO3.

Similar to many other X3 systems, the cyclic N3 radicalfeatures CI in the ground state,12 with unusually strong geo-metric phase effect that changes the nodal structure of thevibrational wave function and the ordering of vibrationallevels.11

Cyclic N3 was described as a metastable molecule tenyears ago,13 which was confirmed by later calculations.14,15

The first experimental evidence of cyclic N3 was obtained byWodtke and co-workers,16–18 who reported the cyclic N3 pro-duction in the ClN3 photodissociation and measured its ion-a"Electronic mail: [email protected]

THE JOURNAL OF CHEMICAL PHYSICS 124, 224309 !2006"

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ization threshold.18 These experiments motivated recent the-oretical studies of N3, as well as the present work targetingthe cyclic N3

+ ground and excited states, in order to facilitatethe interpretation of photoelectron experiments. In contrastto the neutral, only limited information is available about thecation states.19

The cyclic N3 cation is a closed shell molecule of D3hsymmetry with doubly degenerate HOMO and LUMO. Thus,the lowest excited states of each multiplicity are almost qua-druply degenerate !with two exactly degenerate states" andexhibit JT-like behavior.

A similar quadruple set of the excited states occurs inother systems with doubly degenerate HOMO and LUMO,for example, in benzene,20 a rather popular JT system. Inter-estingly enough, most theoretical studies of the JT effects inbenzene were focused on the benzene cation, which exhibitsthe usual JT conical intersection. As for the neutral molecule,most of the computational studies reported only the verticalexcitation energies21,22 for these four almost degenerate ex-cited states.

The effect of other closely lying electronic states on JTintersections has been discussed by several researchers.23–25

For example, Perrin and Gouterman23 analyzed the !E+A"! e vibronic coupling problem. In their treatment, degenerateE states that form CI were considered not as isolated states!e.g., as in the E ! e vibronic coupling problem26" but asbeing coupled to a closely lying A state. Similar interactionswere characterized in Na3,

24 where the interpretation of theexperimental data required the inclusion of the nondegener-ate A1 state and treating all three states as a pseudo-Jahn-Teller system.

This work presents a comprehensive analysis of the cy-clic N3

+ excited states. We found that two HOMO→LUMOexactly degenerate states do not form a familiar A2 /B1 coni-cal intersection because of the presence of the two otherclosely lying HOMO→LUMO states of the A2 and B1 sym-metries. Nondegenerate states are coupled to the degenerateones forming an !E+A+B" ! e vibronic problem, and theenergy dependence on the displacement from the intersectionbecomes purely quadratic, except for the points of an acci-dental degeneracy of three states. Such glancing intersectionis similar to Renner-Teller glancing intersections in linearmolecules.27 It exhibits pseudo-Jahn-Teller distortions, as op-posed to CI and the usual JT effect characterized by thelinear dependence of the energy along the displacements, andhas no geometric phase effect in the electronic wavefunctions.7

The structure of the paper is as follows. The next sectiondescribes computational details. Molecular orbital frame-work and the nature of low-lying states is presented in Secs.III and IV, respectively. The formal analysis of the !E+A+B" ! e JT problem is presented in Sec. V. Section VI dis-cusses ionization potential and photoelectron spectrum. Ourfinal remarks are given in Sec. VII.

II. COMPUTATIONAL DETAILS

Excited state equilibrium geometries, frequencies, andvertical and adiabatic excitation energies were calculated at

the EOM-CCSD !Refs. 28 and 29"/cc-pVTZ !Ref. 30" levelof theory with frozen 1s core orbitals. Potential energy sur-faces of the excited states were obtained at the EOM-CCSDlevel with the 6-311G* !Ref. 31" basis set. EOM-CCSD/cc-pVTZ PESs presented in this work will be discussed in moredetail elsewhere.32 The ground state of cyclic N3

+ was char-acterized using the CCSD model with perturbative triplescorrections, CCSD!T",33 and the cc-pVTZ !Ref. 30" basis set.

The ionization potential of cyclic N3 was calculated asthe energy difference between the neutral and the cation us-ing the CCSD!T" total energies with the following bases:30

cc-pVDZ→cc-pVTZ→cc-pVQZ→cc-pV5Z !1s core orbit-als were frozen". The three latter basis sets were used for thethree point basis set extrapolation CBS-3pa !Refs. 34 and 35"of the neutral and cation total energies,

Ecc-pVXZ!X" = EBSL + be−cX,

where Ecc-pVXZ is the total energy obtained with cc-pVXZbasis set, X denoting a cardinal number: X= %T,Q,5&, EBSL isthe extrapolated basis set limit energy, and b and c are thefitting constants. The EOM-SF-CCSD !Refs. 36 and 37"/cc-pVTZ equilibrium geometry of the neutral cyclic N3 and theCCSD!T"/cc-pVTZ equilibrium geometry of the cation wereused in the IP calculations. The zero point energy !ZPE" ofneutral cyclic N3 including the geometric phase effect isfrom Ref. 11. Cation’s ZPE is calculated at the harmonicapproximation at the CCSD!T"/cc-pVTZ level of theory.

EOM-EE-CCSD and EOM-SF-CCSD results were ob-tained with the Q-CHEM !Ref. 38" ab initio package.CCSD!T" calculations were performed with the ACES II!Ref. 39" electronic structure program.

III. MOLECULAR ORBITAL PICTURE

The ground state equilibrium geometry of cyclic N3+ is an

equilateral triangle !D3h" with RNN=1.313 Å. Figure 1 shows

FIG. 1. Molecular orbitals and the ground state electronic configuration ofcyclic N3

+ !equilateral triangle, D3h". Both HOMO and LUMO are doublydegenerate. C2v labels are given in parentheses.

224309-2 Mozhayskiy, Babikov, and Krylov J. Chem. Phys. 124, 224309 "2006!


MOs and the ground sate electronic configuration of N3+,

which is a closed shell molecule with A1! !A1 in C2v" elec-tronic wave function. MOs are derived from: !i" the sp2 hy-bridized 2s, 2px, and 2py atomic orbitals, which form ninemolecular orbitals—three !-bonding, three !-antibonding!!*", and three lone pair !lp" orbitals, and !ii" 2pz atomicorbitals that form three "-like MOs. Each triple set of MOs!i.e., !, !*, lp, or "" exhibits a similar pattern—one fullybonding MO lies bellow two exactly degenerate orbitals.There is no clear energy separation between the " and lpsets. An interesting feature of this molecule is that bothHOMO and LUMO are doubly degenerate. HOMOs !lp’s"are of e! symmetry, whereas LUMOs !"*" are of e" symme-try. At C2v, the HOMO pair splits into a1 and b2, andLUMO—into the a2 and b1 orbitals.

IV. LOWEST EXCITED STATES

The least symmetric configuration of a triangular mol-ecule is Cs. We use C2v symmetry labels for the twelve low-est excited states, which all are of either A2 or B1 symmetryand therefore become A" at Cs. D3h labels are also givenwhen appropriate.

Figure 2 shows leading configurations of the ground andthe lowest excited states. Different determinants are com-bined in configuration state functions !CSFs" that have ap-propriate spin and spatial symmetry and represent a conve-nient basis for describing excited states.

The lowest excited states of cyclic N3+ are derived from

the eight possible single excitations from doubly degenerateHOMO to doubly degenerate LUMO !lp→"*, top panel".The symmetries of respective CSFs are given by !a1+b2"! !a2+b1"= !A2+A2+B1+B1", and each of these can be ei-ther a singlet or a triplet. Thus, a total of eight different CSFscan be formed, as shown in Fig. 2 !lower panel". Labels #, $,%, and & denote different types of CSF, whereas minus orplus signs correspond to singlet or triplet configurations, re-

spectively. Next two singlets and triplets !' and (" are de-rived from the excitations from nondegenerate HOMO-2 toLUMO.

Only CSFs of the same spin and irrep can mix in theexcited state wave functions. For example, two singlet A2states are described as linear combinations of two singlet A2CSFs,

'!# + &" 1A2( =1

))2 + *2 #) · '!#"1A2( + * · '!&"

1A2($

'!# − &" 1A2( =1

))2 + *2 #* · '!#"1A2( − ) · '!&"

1A2($ !1"

The wave functions of two triplet '!#±&" 3A2( states canbe formed in a similar way. Likewise, four B1 states !twosinglets and two triplets" '!$±%"B1( are linear combinationsof '!$"B1( and '!%"B1( CSFs. The coefficients ) and * areboth equal to one at D3h, whereas along C2v distortions !e.g.,along the bending normal mode" their ratio changes. In otherwords, at D3h the excited state wave functions are simply asum or a difference of basis CSFs, and a C2v distortion col-lapses each state into single CSF !see Fig. 3". Thus, theseCSFs represent an approximate diabatic basis. Overall, atD3h geometries four singlet and four triplet excited statewave functions are formed from four CSFs with *=)=1:'!#+&" 1,3A2(, '!#−&"

1,3A2(, '!$+%"1,3B1(, and '!$

−%" 1,3B1(. Since all CSFs are derived from the single exci-tations between the two pairs of doubly degenerate orbitals,the resulting states are also nearly degenerate and form arather complicated D3h intersection. The '!'" 1,3A2( and'!(" 1,3B1( states derived from HOMO-2→LUMO excita-tions do not mix at D3h geometry and form a regular JT pair.

Using D3h symmetry labels, the symmetries of theHOMO→LUMO states are

e! ! e" → A1" + A2" + !E"" ——→C2v

A2 + B1 + !A2 + B1" ,!2"

i.e., among these four excited states only one A2 and one B1state are exactly degenerate E" states forming a JT pair.

FIG. 2. Leading electronic configurations of the ground !top panel" and thelowest excited states !lower panel". HOMO→LUMO excitations give riseto four singlet and four triplet CSFs labeled #, $, %, and &. HOMO-2→LUMO excitations yield two additional CSFs of each multiplicity: !'"and !(".

FIG. 3. Changes in the excited states’ characters and potential energy sur-faces upon distortions from an equilateral !D3h" geometry to an obtuse !left"and an acute !right" isosceles C2v triangles.

224309-3 Cyclic trinitrogen cation J. Chem. Phys. 124, 224309 "2006!


These degenerate states are '!#−&"A2( and '!$+%"B1(.HOMO-2→LUMO states, !'" 1,3A2 and !("

1,3B1, form an-other E! JT pair. Calculated vertical excitation energies forthe 12 lowest excited states are summarized in Table I. Asexplained above, only two of the four HOMO-LUMO statesare exactly degenerate; however, the order of the states isdifferent for singlets and triplets.

PES scans along the bending normal coordinate areshown in Fig. 4 separately for each irrep and multiplicity.The coordinate origin !Qb=0.0" corresponds to N3

+ at theequilateral D3h geometry !RNN=1.313 Å", whereas left andright wings of the plots correspond to C2v distortions. Thescale along the bending normal mode is as follows: Qb=0.4 corresponds to +=45.3° and RNN=1.430 Å, whereasQb=−0.4—to +=77.2° and RNN=1.220 Å.

The ground X 1A1 state, which has D3h equilibrium ge-

ometry, is shown by the solid line and empty circles on eachplot. The lowest excited states !filled squares, triangles, andcircles" have singly occupied degenerate orbitals and un-dergo JT distortions to C2v.

Let us first discuss the !'" 1,3A2 and !("1,3B1 states de-

rived from the excitations from nondegenerate HOMO-2 todoubly degenerate LUMO !two lowest CSFs in Fig. 2".These states are shown in the each plot in Fig. 4 by the verytop dashed line. The two singlets, !'" 1A2 and !("

1B1, areexactly degenerate at D3h and undergo JT distortions to C2v.The stereographic projections of PESs using hypersphericalcoordinates40 of PESs shown in Fig. 5 reveal the similarity ofthis intersection to the 2B1 /

2A2 intersection in neutral cyclicN3.

11 Transition from the acute to the obtuse triangle station-ary points through the D3h point !i.e., along the bending nor-mal mode" encounters a relatively high potential barrier;however, the molecule can go around the conical intersectionwith almost no barrier following the asymmetric normalmode that corresponds to pseudorotation. The seam of thisconical intersection is along the fully symmetric stretch. Thepair of the !'"3A2 and !("

3B1 triplet states follows the samepattern, although the energy differences between the respec-tive equilibrium geometries !EG" and the transition states!TS" are slightly larger !see Table II", and the barrier forpseudorotation is higher. Such A2 /B1 intersection causes theelectronic wave function to gain a phase !change of a sign"along any path on the adiabatic PES that encircles CI.11

The four lower excited states !#, $, %, and &" show adifferent and more complicated behavior, due to the doubledegeneracy of initial and target MOs !Fig. 2". Instead of a JTpair, they form a JT quartet: four almost degenerate elec-tronic states, which are all unstable at D3h and distort tolower symmetries. The calculated PES of this intersection ispresented in Fig. 6, and the excited state characters aroundthe intersection point are sketched in Fig. 3 !see also PES

TABLE I. Vertical excitation energies !eV" of the 12 lowest excited states ofcyclic N3

+ calculated at the EOM-CCSD/cc-pVTZ level of theory. Two out offour HOMO→LUMO !#−&, $+%" and two HOMO-2→LUMO !', ("excited states are exactly degenerate pairs at D3h. The ground state geometryand total energy are given in Table II.

Singlets Triplets

!(" 1A2 7.669 !("3A2 7.014

!'" 1B1 7.669 !'"3B1 7.014

!#−&" 1A2 5.334 !#+&"3A2 4.333

!$+%" 1B1 5.334!$−%" 3B1 3.950

!$−%" 1B1 5.314!$+%" 3B1 3.921

!#+&" 1A2 4.914 !#−&"3A2 3.921

FIG. 4. The EOM-CCSD/6-311G* potential energy surface scans along thebending normal coordinate for the ground !X 1A1, shown on each plot" andthe excited 1A2,

1B1,3A2,

3B1 states of cyclic N3+ !upper left, upper right,

lower left, and lower right, respectively". Data points !squares, triangles, andfilled circles" correspond to the calculated adiabatic surfaces; dashed linesrepresent approximate diabats and connect points with the same leadingcharacter of the wave function !see Fig. 2".

FIG. 5. Adiabatic potential energy surfaces and contour plots of the !'"A2and !("B1 singlet !left" and triplet !right" states. Polar radius and angle arehyperspherical coordinates + and ,, which are similar to the bending andasymmetric stretch normal mode, respectively. Stereographic projection istaken with fixed hyperradius !overall molecular size or symmetric stretch"-=3.262 corresponding to the cyclic N3

+ ground state equilibrium geometry.Both surfaces feature conical !'"A2 / !("B1 intersection at D3h. Stationarypoints on the surfaces are located along two C2v distortions to acute andobtuse isosceles triangles. Energies of the transition state and the conicalintersection relative to the minima are ETS=0.05 eV, ECI=0.97 eV andETS=0.24 eV, ECI=1.03 eV on the singlet and tiplet !' /(" PESs,respectively.



cuts along C2v distortions in Fig. 4". Obviously, this is not aquadruply degenerate intersection—only two out of the fourelectronic states are exactly degenerate at D3h, as describedabove. Nevertheless, all four states around the intersectionstrongly interact, which results in a fascinating pattern. Forexample, the intersection of the two degenerate states isglancing rather than conical, and the energy depends onlyquadratically on the displacements from the intersectionpoint,6,7 as explained in detail in Sec. V. Note that the sin-glets !left panel, Fig. 3" form almost a triply degenerate in-tersection: !$−%" 1B1 state is only 0.02 eV !Table I" lowerthan the !#−&" 1A2 / !$+%"

3B1 pair. Interestingly, the orderand the character of the excited states at D3h is different forsinglets and triplets, although it becomes the same for thegeometries distant from the equilateral triangle as shown bythe labels on the left and right sides of the plots in Fig. 3.

The intersection topology is further clarified by the ad-ditional scan along the fully symmetric stretch normal bondQss that corresponds to changing the overall size of the equi-lateral triangle !Fig. 7". All four states of each multiplicityremain close in energy along Qss, and the two exactly degen-erate states retain their degeneracy !solid line in Fig. 7" form-ing the seam of the glancing intersection. However, the non-degenerate states !filled circles and squares" slightly changetheir energy position relative to the glancing intersection,which changes the order of the triplet states and even leads toadditional accidental degeneracies of the !#+&" 3A2 or !$−%" 3B1 states with the glancing intersection at some D3hgeometries !encircled in Fig. 7".

Optimized geometries, frequencies, and adiabatic excita-

tion energies of the excited states described above are pre-sented in Table II. As in many JT triatomics, one sheet of!$" / !%" singlet and triplet PESs has a minimum !EG", andanother—a transition state, the later corresponding to the po-tential barrier for pseudorotation motion between the equiva-lent EG minima. !#" / !&" states could follow the similar pat-tern; however, the !#" singlet and triplet states aredissociative along the C2v distortion. If the C2v constrain islifted, the !#" states relax to linear structures: the !#" 1A2assumes C.v geometry with RNN=1.178 Å, whereas the trip-let !#" 3A2 becomes D.h with R12=1.123 Å and R23=1.279 Å.

The shape of the # /& and $ /% PES crossings is differentfrom that of ' /( CI !and also from the 2B1 /

2A2 intersectionin neutral cyclic N3", as the surfaces # /& and $ /% have onlya quadratic dependence on a displacement from the intersec-tion point. Thus, even though there are two exactly degener-ate JT states, the intersection is glancing rather than conicaland adiabatic PESs do not have singularities.

This changes the behavior of electronic wave function inthe vicinity of the intersection: although the character of theelectronic wave function changes along the path around theintersection, the point group symmetry remains the same,and, therefore, the electronic wave function does not gain asign change. Thus, there is no geometric phase effect alongany path that stays on one of the four adiabatic PESs andencircles this intersection,6,7 which is not surprizing, in viewof the absence of a singularity point.

TABLE II. C2v constrained optimized geometries, harmonic vibrational frequencies, and total !Etot" and adiabatic excitation !Eex" energies of the ground!X 1A1" and the lowest excited states calculated at the EOM-CCSD/cc-pVTZ level of theory. /1, /2, and /3 are the frequencies of the symmetric stretch,bending, and asymmetric stretch, respectively.

X 1A1 !#"1A2 !$"

1B1 !%"1B1 !&"

1A2 !'"1A2 !("

1B1

Eex !eV" 0.00 2.92 5.16 5.19 2.51 6.76 6.81Etot !a.u." −163.423 29 −163.315 93 −163.233 79 −163.232 66 −163.331 06 −163.174 96 −163.172 89

+ !°" 60.0 37.7 56.0 62.1 92.2 52.3 66.3RNN !Å" 1.313 1.750 1.389 1.343 1.244 1.475 1.365Enuc !a.u." 59.237 38 55.523 43 57.234 05 57.328 02 56.139 34 55.077 21 55.364 57

/1 !cm−1" 1646 2134 1427 1401 1633/2 !cm−1" 1108 363 823 1013 597 n/a

b n/ab

/3 !cm−1" 1108 −i424 1163 −i1362 764ZPE !kcal/mol" 5.521 3.830a 4.880 3.451a 4.280

!#" 3A2 !$"3B1 !%"

3B1 !&"3A2 !'"

3A2 !("3B1

Eex !eV" 1.15 3.85 3.82 1.94 6.04 6.28Etot !a.u." −163.380 90 −163.281 92 −163.283 07 −163.352 13 −163.201 32 −163.192 52

+ !°" 34.1 57.0 63.3 90.9 51.9 64.4RNN !Å" 1.910 1.360 1.322 1.249 1.473 1.372Enuc !a.u." 50.334 01 58.090 07 57.906 82 56.091 25 55.332 35 55.522 38

/1 !cm−1" 2241 1429 1419 1603 2701 1751/2 !cm−1" 434 889 999 599 1129 1361/3 !cm−1" −i385 −i1010 1470 −i1366 1514 −i1669ZPE, kcal/mol 3.570a 3.313a 5.558 3.148a 7.639 4.450a

aZPE for the transition states were calculated only for normal coordinates with real frequencies, i.e., bending and symmetric stretch.bWe were not able to calculate frequencies for the !'" and !(" states because of the numerical instability of finite difference procedure in the vicinity of theconical intersection.



V. THE ANALYSIS OF THE „E+A+B…‹E PROBLEM INCYCLIC N3

+

The electronic Hamiltonian H=Te+U!r ,Q" can be ex-panded as Taylor series with respect to small nuclear dis-placements Q- from a reference high symmetry configura-tion !Q-=0",

H = H0 + *-

#H

#Q-Q- + *

-,!

#2H

#Q-#Q!Q-Q! + ¼ = H0 + V .

!3"

We truncate this expansion at linear terms and start bysolving the Schrödinger equation for Hamiltonian H0. Theperturbation V thus includes linear vibronic coupling terms*-!#H /#Q-"Q-.

1,2

Instead of taking eigenfunctions of H0 as a basis set fora subsequent perturbative treatment, we choose to employ adiabatic basis of HUMO→LUMO CSFs !see Fig. 2". TheseCSFs are close to adiabatic states for C2v distorted geom-etries, whereas at D3h !Q-=0" the corresponding adiabaticstates !i.e., eigenstates of H0" are the linear combinations ofCSFs, as given by Eq. !1".

We employ normal coordinates: bending Qb, asymmetricstretch Qas and symmetric stretch Qss, which are of a1, b2,and a1 symmetry !in C2v", respectively. We will consider Qband Qas, which constitute the e! degenerate vibration. Thethird normal coordinate Qss describes breathing motion,which does not lift the degeneracy between MOs and CSFs.This mode will be discussed in the end of the section.

The matrix elements Vij of the vibronic coupling termare

Vij = +0i'*-

#H

#Q-Q-'0 j( = *

-

+0i'#U

#Q-'0 j(Q-

= *-

FijQ-Q-, !4"

where %0k& are the diabatic %!#"A2 , !&"A2 , !$"B1 , !%"B1& ba-sis functions.

Selection rules for FijQ-,41 derivative or linear vibronic

coupling constant, are readily derived from the group theoryconsiderations. Vij is nonzero only if 1+i' ! 1Q- ! 1+'j( includestotally symmetric irrep A1, where 1+i', 1'j(, and 1Q- are theirreps of the 0i, 0 j diabats and the Q- normal mode, respec-tively. Thus, the linear vibronic coupling is nonzero betweenthe states of the same symmetry only along the bending nor-mal coordinate, e.g., 1+B1' ! 1Qb!a1" ! 1+'B1(!A1. For the statesof different symmetry, i.e., A2 and B1, it is nonzero onlyalong the asymmetric stretch: 1+A2' ! 1Qas!b2" ! 1+'B1(!A1.Thus, the vibronic coupling matrix elements Vij are

+0iA2'V'0 j

A2( = FijQbQb,

+0iB1'V'0 j

B1( = FijQbQb, !5"

+0iB1'V'0 j

A2( = FijQasQas.

The H0 off-diagonal matrix elements are nonzero onlybetween the states of the same symmetry,

FIG. 6. !Color" PESs of the ground !X" and the first eight excited states ofcyclic N3

+. Coordinates are as in Fig. 5. Three out of four states in eachmultiplicity are almost degenerate at D3h geometry, two being exactlydegenerate.

FIG. 7. EOM-CCSD/6-311G* potential energy surfacescans along symmetric stretch normal coordiante for thelowest A2 and B1 excited states. Singlets are shown onthe left plot, triplets—on the right. Solid line shows twoexactly degenerate states the !#−&"A2 and !$+%"B1,i.e., the seam of the intersection. Circles and squarescorrespond to the nondegenerate !#+&"A2 and !$−%"B1 states, respectively. Big circles on the right plotshow two tree-state PES intersections. RNN is a bondlength of equilateral triangle, vertical dashed line pointsat the cyclic N3

+ ground state equilibrium geometry.



+0iA2'H0'0 j

A2( = VAA0 ,

+0iB1'H0'0 j

B1( = VBB0 , !6"

+0iB1'H0'0 j

A2( = 0.

Using Eqs. !5" and !6", the Hamiltonian in the diabaticbasis set %!#"A2 , !&"A2 , !$"B1 , !%"B1& assumes the followingform:

H!Qb,Qas"

=,EA + kAQb VAA

0 F#$Qas F#%QasVAA

0 EA − kAQb F&$Qas F&%QasF#$Qas F&$Qas EB + kBQb VBB

0

F#%Qas F&%Qas VBB0 EB − kBQb

- ,!7"

where kA=F##Qb =−F&&

Qb and kB=F$$Qb =−F%%

Qb.Along the bending normal mode Qb, when Qas=0, CSFs

of different symmetries are not coupled, and the Hamiltonian!7" assumes a block diagonal form. Thus, pairs%!#"A2 , !&"A2& and %!$"B1 , !%"B1& form two pairs of non-crossing adiabats !see Fig. 8, compare to Fig. 3".

At D3h, i.e., when Qb=0 and Qas=0, it is required bysymmetry, Eq. !2", that one of the A2 states !UA

±" is degener-ate with one of the B1 states !UB

±", i.e., in the example shownin Fig. 8 the intersection condition is UA

+ =UB+, which gives

rise to the additional condition EA+ 'VAA0 '=EB+ 'VBB

0 '. Byshifting the energy scale such that one of the diabatic ener-gies at the intersection is zero, this condition becomes

'VBB0 ' = 'VAA

0 ' − EB,!8"

EA = 0.

The coupling between the two degenerate states at !Qb=0, Qas=0" is zero by virtue of Eqs. !5" and !6". Thus, thederivatives of the potential energy surfaces !Ui" along thebending coordinate !see Fig. 8" are zero,

. #Ui#Qb

.Qas=Qb=0

= 0, i = 1¼4, !9"

which means that the linear terms are absent and the inter-section is glancing rather than conical.

The Hamiltonian along the asymmetric stretch Qas is ob-tained from matrix !7" by using Qb=0 and condition !8". Ifonly two intersecting states are considered, the problem issimilar to the familiar conical intersection4,5 of E" degeneratestates,

H = / kAQb F#$QasF#$Qas kBQb

0 . !10"However, because of the two other states A2 and B1,

which are almost degenerate with E" pair, the linear vibroniccoupling constants Fij

Qas are nonzero and the 424 full Hamil-tonian should be considered at the first order of perturbationtheory.

This 424 problem can be solved analytically, e.g., byusing MATHEMATICA.42 The resulting !rather tedious!" ex-pressions for eigenvalues can be differentiated, which revealsthat the eigenvalues’ derivatives along the asymmetricstretch coordinate are also zero, similar to the derivativesalong Qb. Thus,

. #Ui#Qas

.Qas=Qb=0

= 0, i = 1¼4, !11"

and therefore all four potential energy surfaces !E"+A2+B1" depend only quadratically on the displacements alongthe degenerate vibration e"=Qb+Qas from D3h geometry, i.e.,they have an extremum at the symmetric configuration. Notethat the inclusion of the second order terms in the perturba-tion does not change the derivatives in Eqs. !9" and !11".

Thus, the intersection of four HOMO-LUMO excitedstates is glancing,43–45 and all four !E+A+B" ! e vibroni-cally coupled states follow a pseudo-Jahn-Teller distortion.Contrary to the conical intersection case, there is no geomet-ric phase effect along any path that encircles the intersectionpoint.7,24,26

The absence of linear terms can also be demonstrated byconsidering 222 block of the Hamiltonian in the basis ofadiabatic JT states !i.e., two degenerate eigenstates of H0", aselegantly shown by Pupyshev.46 The proof requires the con-struction of complex e! and e" MOs obtained from a1 /b2HOMOs and a2 /b1 LUMOs,

FIG. 8. The Hamiltonian in the diabatic !left" and adiabatic !right" repre-sentations along the bending normal mode Qb. Since the Hamiltonian isblock diagonal, the pairs of states of the same symmetry do not interact witheach other and form two noncrossing pairs !see text".



e±! =1)2 !a1 ± ib2" ,

!12"

e±" =1)2 !− b1 ± ia2" .

In this basis, two degenerate adiabatic states 0E+ and0E− are simply single e+!→e−" and e−!→e+" excitations,

01,3E+=

1)2 !'e+!# e−!# e−!$ e−"$( ± 'e+!$ e−!# e−!$ e−"#(" ,

!13"

01,3E−=

1)2 !'e+!# e+!$ e−!# e+"$( ± 'e+!# e+!$ e−!$ e+"#(" .

The above can be derived by either transforming the originaladiabatic states into the new MO basis or by symmetry con-siderations. In terms of diabatic CSFs from Fig. 2, thesestates are

01,3E±=

1)2 #!'!#"A2( − '!&"A2(" ± i!'!$"B1( + '!%"B1("$ .

!14"

As clearly seen from Eq. !13", 0E+ is doubly excitedwith respect to 0E−. Neglecting the changes in MOs uponsmall geometric distortion, the perturbation operator #U /#Qis a one-particle operator, and, therefore, the correspondingmatrix element is zero. There is no linear dependence in+0E±'#U /#Qas'0E±( and +0E±'#U /#Qb'0E±( diagonal termsbecause of the symmetry.46 Note that the double degeneracyof both initial and target MOs !e.g., HOMO and LUMO" isrequired for the two respective electronic states to be doublyexcited with respect to each other. Thus, both proofs showthat the cancellation of linear terms occurs due to the pres-ence for of four interacting CSFs.

The change of the overall size of cyclic N3+ under D3h

constraint corresponds to the symmetric stretch !trianglebreathing" Qss !a1" motion, with Qb=Qas=0. The E" or A2+B1 pair of states remains degenerate at any point along Qss.Both zero and first order coupling terms between the statesof different symmetries are identically zero by symmetry,which means that the A2 states do not interact with the B1states along Qss, as well as along the Qb normal mode of thesame a1 symmetry. However, the derivative coupling be-tween two states of the same symmetry, e.g., '!#"A2( and'!&"A2(, is nonzero along the Qss and can accidentally cancelout the zero order coupling term VAA, which results in a tripledegeneracy !E"+A2". By setting Qss to be equal to zero atsuch triple degeneracy point and taking into account condi-tions !8", Hamiltonian !7" assumes the following form:

, kAQb0 F#$Qas F#%Qas

0 − kAQb F&$Qas F&%QasF#$Qas F&$Qas EB + kBQb − EBF#%Qas F&%Qas − EB EB − kBQb

- . !15"

In this case, both derivatives #Ui /#Qb and #Ui /#Qas fori=1¼4 at Qas=Qb=0 are nonzero, and the intersection has aconical shape in !Qas ,Qb" coordinates: triple conical inter-section and a nondegenerate fourth surface with the singular-ity in the origin !Qas=0, Qb=0". Triple CIs, which are notdefined by the high nuclear symmetry, were characterized byMatsika and Yarkony47,48 as an accidental intersection of twoseams of conical intersections. In triatomics, this type of CIwas found, for example, in H2+H.

49 In cyclic N3+, however,

the triple CI is formed by two crossing seams of A" /A" coni-cal intersection !in Cs" and the A2 /B1 glancing intersection!along D3h".

To conclude, interactions of a JT pair of states with otherstates remove linear terms and change the intersection fromconical to glancing thus eliminating geometric phase effects.

VI. IONIZATION POTENTIAL AND PHOTOELECTRONSPECTRUM

The cyclic N3+ ionization potential was calculated by

CCSD!T" with the basis set extrapolation, as described inSec. II. The results are presented in Fig. 9. Energies of boththe neutral and the cation were calculated with four differentbasis sets. The energy difference extrapolated to the basis setlimit is 10.52 eV. The zero point energy !ZPE" of the cationin the harmonic approximation is 0.239 eV !see Table II",and for the neutral cyclic N3 E-symmetry vibrational groundstate—0.164 eV.11 Thus, with the ZPE correction, the adia-batic IP00 is 10.595 eV. This result supports the experimentalmeasurement of IP for cyclic N3

+ !Ref. 18" of10.62±0.07 eV.

The IP of the linear N3 radical is 11.06 eV.50 Therefore,

the difference in IPs of the cyclic and linear N3 is only0.44 eV, which is comparable to the vibrational energy ofpossibly hot photofragments. Thus, a more detailed analysisof photoelectrons is required to unambiguously assign theobserved product as cyclic N3. Below we discuss generalfeatures of the cyclic N3 photoelectron spectrum. The calcu-lation of the spectrum and the comparison with the experi-ments will be reported in a forthcoming paper.32

FIG. 9. IPee !not ZPE corrected" calculated as a difference between neutral’sand cation’s CCSD!T" total energies in the basis set limit. ZPE corrected IP,IP00, is 10.595 eV.



The lowest electronic states of the cation that are brightin a photoelectron experiment are those that are derived fromneutral cyclic N3 by one electron ionization: the groundX 1A1 and !#", !$", !(" excited states, as well as !%", !&", and!'" for the photoionization from 2B1 !EG" and

2A2 !TS",respectively. Since neutral’s vibrational wave function is de-localized alonog the pseudorotation coordinate over 2B1 and2A2 states,

12 both sets of the cation’s excited states can beproduced in one electron photoionization.

Thus, all lowest excited states discussed in this work cancontribute to the photoelectron spectrum, and, since they areclose in energy and strongly coupled, the calculation of thefull photoelectron spectrum becomes a challenging problem.

The Franck-Condon factors, however, are very differentfor many of the states. For example, the lowest excitedstates, !#" 1A2 and !#"

3A2, are almost dissociative along C2vdistortion !see Table II" and collapse to linear equilibriumstructures. Because of this geometry difference, the Franck-Condon factors for these # states and neutral N3 are small,and they should produce only a background signal in thephotoelectron spectrum. All other excited states are at least3.8 eV higher than the cation ground state. Thus, the lowerenergy part !Eex314 eV" of the photoelectron spectrum canbe well described as a transition from the 2B1 /

2A2 pair ofstates of the neutral to the ground X 1A1 state of the cation.Such calculation should include the geometric phase effect inneutral N3 and will be presented elsewere.

32

VII. CONCLUSIONS

We described 12 lowest excited states of cyclic N3+. Eight

lowest states !four singlets and four triplets" derived fromsingle electronic excitations from doubly degenerate HOMOto doubly degenerate LUMO are close in energy at theground state equilibrium geometry !D3h" and exhibit a com-plicated Jahn-Teller behavior. Only two out of four states ineach multiplicity are exactly degenerate and form an inter-section seam along the symmetric stretch normal mode.However, this intersection is glancing rather than conical,because it is affected by interactions with two other nonde-generate states. Thus, adiabatic PESs do not have singulari-ties at the intersections point, unless accidental triple degen-eracy occurs. Therefore, these glancing intersections do notcause a geometric phase effect, which occurs in cyclic N3ground state or any other system with CI.

Stationary points of the excited states PES were alsocharacterized. Cyclic N3 ionization potential was estimatedto be 10.595 eV, in a good agreement with recent experi-ments. The photoelectron spectrum of cyclic N3 in the en-ergy range Eex314 eV is predicted to be dominated by thetransitions between the lowest electronic states of neutral N3and the ground state of the cation.

ACKNOWLEDGMENTS

We are very grateful to Dr. Vladimir I. Pupyshev !Mos-cow State" for critical and insightful comments. We wouldalso like to thank Alec Wodtke !UCSB" for helpful discus-sions and providing the experimental data. University ofSouthern California Center for High Performance Comput-

ing and Communications for making their computational re-sources available. A.I.K. acknowledges the support from theDepartment of Energy !DE-FG02-05ER15685".

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conical and glancing jahn-t eller intersections in the cyclic … · 2019. 5. 21. · conical and...

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